Average Error: 14.8 → 14.3
Time: 13.4s
Precision: 64
\[1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{\frac{\frac{\frac{1}{512} - \frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right)} \cdot \left(\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right)} \cdot \frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right)}\right)}{\left(\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right)} + \frac{1}{8}\right) \cdot \frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right)} + \frac{1}{64}}}{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) + \frac{1}{4}}}{\sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}} + 1}\]
1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{\frac{\frac{\frac{1}{512} - \frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right)} \cdot \left(\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right)} \cdot \frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right)}\right)}{\left(\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right)} + \frac{1}{8}\right) \cdot \frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right)} + \frac{1}{64}}}{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) + \frac{1}{4}}}{\sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}} + 1}
double f(double x) {
        double r4612350 = 1.0;
        double r4612351 = 0.5;
        double r4612352 = x;
        double r4612353 = hypot(r4612350, r4612352);
        double r4612354 = r4612350 / r4612353;
        double r4612355 = r4612350 + r4612354;
        double r4612356 = r4612351 * r4612355;
        double r4612357 = sqrt(r4612356);
        double r4612358 = r4612350 - r4612357;
        return r4612358;
}
↓
double f(double x) {
        double r4612359 = 0.001953125;
        double r4612360 = 0.125;
        double r4612361 = 1.0;
        double r4612362 = x;
        double r4612363 = hypot(r4612361, r4612362);
        double r4612364 = r4612363 * r4612363;
        double r4612365 = r4612363 * r4612364;
        double r4612366 = r4612360 / r4612365;
        double r4612367 = r4612366 * r4612366;
        double r4612368 = r4612366 * r4612367;
        double r4612369 = r4612359 - r4612368;
        double r4612370 = r4612366 + r4612360;
        double r4612371 = r4612370 * r4612366;
        double r4612372 = 0.015625;
        double r4612373 = r4612371 + r4612372;
        double r4612374 = r4612369 / r4612373;
        double r4612375 = 0.5;
        double r4612376 = r4612375 / r4612363;
        double r4612377 = r4612375 + r4612376;
        double r4612378 = r4612376 * r4612377;
        double r4612379 = 0.25;
        double r4612380 = r4612378 + r4612379;
        double r4612381 = r4612374 / r4612380;
        double r4612382 = sqrt(r4612377);
        double r4612383 = r4612382 + r4612361;
        double r4612384 = r4612381 / r4612383;
        return r4612384;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.8

    \[1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]

  2. Simplified14.8

    \[\leadsto \color{blue}{1 - \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]
  3. Using strategy rm

  4. Applied flip--14.8

    \[\leadsto \color{blue}{\frac{1 \cdot 1 - \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}} \cdot \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}}\]

  5. Simplified14.3

    \[\leadsto \frac{\color{blue}{\frac{1}{2} - \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]
  6. Using strategy rm

  7. Applied flip3--14.3

    \[\leadsto \frac{\color{blue}{\frac{{\frac{1}{2}}^{3} - {\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{\frac{1}{2} \cdot \frac{1}{2} + \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)}}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]

  8. Simplified14.3

    \[\leadsto \frac{\frac{\color{blue}{\frac{1}{8} - \frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}}{\frac{1}{2} \cdot \frac{1}{2} + \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]

  9. Simplified14.3

    \[\leadsto \frac{\frac{\frac{1}{8} - \frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}{\color{blue}{\frac{1}{4} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)}}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]
  10. Using strategy rm

  11. Applied flip3--14.3

    \[\leadsto \frac{\frac{\color{blue}{\frac{{\frac{1}{8}}^{3} - {\left(\frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{\frac{1}{8} \cdot \frac{1}{8} + \left(\frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{8} \cdot \frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}\right)}}}{\frac{1}{4} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]

  12. Simplified14.3

    \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{1}{512} - \left(\frac{\frac{1}{8}}{\left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right) \cdot \mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{8}}{\left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right) \cdot \mathsf{hypot}\left(1, x\right)}\right) \cdot \frac{\frac{1}{8}}{\left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right) \cdot \mathsf{hypot}\left(1, x\right)}}}{\frac{1}{8} \cdot \frac{1}{8} + \left(\frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{8} \cdot \frac{\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}\right)}}{\frac{1}{4} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]

  13. Simplified14.3

    \[\leadsto \frac{\frac{\frac{\frac{1}{512} - \left(\frac{\frac{1}{8}}{\left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right) \cdot \mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{8}}{\left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right) \cdot \mathsf{hypot}\left(1, x\right)}\right) \cdot \frac{\frac{1}{8}}{\left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right) \cdot \mathsf{hypot}\left(1, x\right)}}{\color{blue}{\left(\frac{1}{8} + \frac{\frac{1}{8}}{\left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right) \cdot \mathsf{hypot}\left(1, x\right)}\right) \cdot \frac{\frac{1}{8}}{\left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right) \cdot \mathsf{hypot}\left(1, x\right)} + \frac{1}{64}}}}{\frac{1}{4} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]

  14. Final simplification14.3

    \[\leadsto \frac{\frac{\frac{\frac{1}{512} - \frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right)} \cdot \left(\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right)} \cdot \frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right)}\right)}{\left(\frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right)} + \frac{1}{8}\right) \cdot \frac{\frac{1}{8}}{\mathsf{hypot}\left(1, x\right) \cdot \left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right)} + \frac{1}{64}}}{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right) + \frac{1}{4}}}{\sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}} + 1}\]

Reproduce

herbie shell --seed 2019153 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  (- 1 (sqrt (* 1/2 (+ 1 (/ 1 (hypot 1 x)))))))